Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6,8,7)(9,10,14,17,13,16,12,21)(11,18,19,15,20,24,22,23);(1,2)(3,4)(5,7,8,6)(9,11,13,20)(10,15,16,23)(12,22,14,19)(17,24,21,18);(2,4)(6,7)(10,16)(15,23)(17,21)(18,24);(9,11)(10,15)(12,22)(13,20)(14,19)(16,23)(17,24)(18,21)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,d^b=di,e^b=ei,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72260037, #72260045, #72262491 |
| maximal normal subgroup of | #72260037, #72260045, #72262491 |
| maximal quotient of | #72258597 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267505);